Question: Simplify; express your answer in exponential form. Assume $x\neq 0, a\neq 0$. $\dfrac{{(x^{-3}a^{3})^{2}}}{{(x^{-1}a^{5})^{-3}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(x^{-3}a^{3})^{2} = (x^{-3})^{2}(a^{3})^{2}}$ On the left, we have ${x^{-3}}$ to the exponent ${2}$ . Now ${-3 \times 2 = -6}$ , so ${(x^{-3})^{2} = x^{-6}}$ Apply the ideas above to simplify the equation. $\dfrac{{(x^{-3}a^{3})^{2}}}{{(x^{-1}a^{5})^{-3}}} = \dfrac{{x^{-6}a^{6}}}{{x^{3}a^{-15}}}$ Break up the equation by variable and simplify. $\dfrac{{x^{-6}a^{6}}}{{x^{3}a^{-15}}} = \dfrac{{x^{-6}}}{{x^{3}}} \cdot \dfrac{{a^{6}}}{{a^{-15}}} = x^{{-6} - {3}} \cdot a^{{6} - {(-15)}} = x^{-9}a^{21}$